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G = C626D4order 288 = 25·32

3rd semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C626D4, C62.120C23, (C2×C6)⋊4D12, C23.28S32, D6⋊Dic38C2, C6.172(S3×D4), C6.87(C2×D12), C34(C127D4), (C3×Dic3)⋊12D4, (C22×C6).78D6, C6.70(C4○D12), Dic35(C3⋊D4), C3214(C4⋊D4), (C22×S3).26D6, Dic3⋊Dic337C2, C34(C23.14D6), C6.D1222C2, C6.57(D42S3), C223(C3⋊D12), (C2×Dic3).105D6, (C22×Dic3)⋊10S3, (C2×C62).39C22, C2.30(D6.3D6), (C6×Dic3).147C22, (C2×C3⋊D4)⋊5S3, (C6×C3⋊D4)⋊11C2, (Dic3×C2×C6)⋊11C2, C2.44(S3×C3⋊D4), C6.24(C2×C3⋊D4), (C2×C6)⋊11(C3⋊D4), C22.143(C2×S32), (C3×C6).166(C2×D4), (C2×C327D4)⋊4C2, (S3×C2×C6).48C22, (C2×C3⋊D12)⋊12C2, (C3×C6).88(C4○D4), C2.25(C2×C3⋊D12), (C2×C6).139(C22×S3), (C22×C3⋊S3).34C22, (C2×C3⋊Dic3).72C22, SmallGroup(288,626)

Series: Derived Chief Lower central Upper central

C1C62 — C626D4
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — C626D4
C32C62 — C626D4
C1C22C23

Generators and relations for C626D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 898 in 215 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×8], S3 [×5], C6 [×6], C6 [×10], C2×C4 [×6], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], Dic3 [×6], C12 [×4], D6 [×13], C2×C6 [×2], C2×C6 [×4], C2×C6 [×12], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3, C3⋊S3, C3×C6 [×3], C3×C6 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×5], C3⋊D4 [×12], C2×C12 [×5], C3×D4 [×2], C22×S3, C22×S3 [×3], C22×C6 [×2], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×3], C2×C3⋊S3 [×3], C62, C62 [×2], C62 [×2], Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C6.D4, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4, C3⋊D12 [×2], C6×Dic3 [×3], C6×Dic3 [×2], C3×C3⋊D4 [×2], C2×C3⋊Dic3, C327D4 [×2], S3×C2×C6, C22×C3⋊S3, C2×C62, C127D4, C23.14D6, D6⋊Dic3, C6.D12, Dic3⋊Dic3, C2×C3⋊D12, Dic3×C2×C6, C6×C3⋊D4, C2×C327D4, C626D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×4], C22×S3 [×2], C4⋊D4, S32, C2×D12, C4○D12, S3×D4, D42S3, C2×C3⋊D4 [×2], C3⋊D12 [×2], C2×S32, C127D4, C23.14D6, D6.3D6, C2×C3⋊D12, S3×C3⋊D4, C626D4

Smallest permutation representation of C626D4
On 48 points
Generators in S48
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 19 5 23 3 21)(2 20 6 24 4 22)(7 26 11 30 9 28)(8 27 12 25 10 29)(13 41 15 37 17 39)(14 42 16 38 18 40)(31 43 33 45 35 47)(32 44 34 46 36 48)
(1 40 10 35)(2 39 11 34)(3 38 12 33)(4 37 7 32)(5 42 8 31)(6 41 9 36)(13 30 46 20)(14 29 47 19)(15 28 48 24)(16 27 43 23)(17 26 44 22)(18 25 45 21)
(1 10)(2 26)(3 8)(4 30)(5 12)(6 28)(7 20)(9 24)(11 22)(13 37)(14 18)(15 41)(17 39)(19 25)(21 29)(23 27)(31 33)(32 46)(34 44)(36 48)(38 42)(45 47)

G:=sub<Sym(48)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,19,5,23,3,21)(2,20,6,24,4,22)(7,26,11,30,9,28)(8,27,12,25,10,29)(13,41,15,37,17,39)(14,42,16,38,18,40)(31,43,33,45,35,47)(32,44,34,46,36,48), (1,40,10,35)(2,39,11,34)(3,38,12,33)(4,37,7,32)(5,42,8,31)(6,41,9,36)(13,30,46,20)(14,29,47,19)(15,28,48,24)(16,27,43,23)(17,26,44,22)(18,25,45,21), (1,10)(2,26)(3,8)(4,30)(5,12)(6,28)(7,20)(9,24)(11,22)(13,37)(14,18)(15,41)(17,39)(19,25)(21,29)(23,27)(31,33)(32,46)(34,44)(36,48)(38,42)(45,47)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,19,5,23,3,21)(2,20,6,24,4,22)(7,26,11,30,9,28)(8,27,12,25,10,29)(13,41,15,37,17,39)(14,42,16,38,18,40)(31,43,33,45,35,47)(32,44,34,46,36,48), (1,40,10,35)(2,39,11,34)(3,38,12,33)(4,37,7,32)(5,42,8,31)(6,41,9,36)(13,30,46,20)(14,29,47,19)(15,28,48,24)(16,27,43,23)(17,26,44,22)(18,25,45,21), (1,10)(2,26)(3,8)(4,30)(5,12)(6,28)(7,20)(9,24)(11,22)(13,37)(14,18)(15,41)(17,39)(19,25)(21,29)(23,27)(31,33)(32,46)(34,44)(36,48)(38,42)(45,47) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,19,5,23,3,21),(2,20,6,24,4,22),(7,26,11,30,9,28),(8,27,12,25,10,29),(13,41,15,37,17,39),(14,42,16,38,18,40),(31,43,33,45,35,47),(32,44,34,46,36,48)], [(1,40,10,35),(2,39,11,34),(3,38,12,33),(4,37,7,32),(5,42,8,31),(6,41,9,36),(13,30,46,20),(14,29,47,19),(15,28,48,24),(16,27,43,23),(17,26,44,22),(18,25,45,21)], [(1,10),(2,26),(3,8),(4,30),(5,12),(6,28),(7,20),(9,24),(11,22),(13,37),(14,18),(15,41),(17,39),(19,25),(21,29),(23,27),(31,33),(32,46),(34,44),(36,48),(38,42),(45,47)])

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6J6K···6S6T6U12A···12H12I12J
order122222223334444446···66···66612···121212
size1111221236224666612362···24···412126···61212

48 irreducible representations

dim111111112222222222224444444
type++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4C3⋊D4D12C3⋊D4C4○D12S32S3×D4D42S3C3⋊D12C2×S32D6.3D6S3×C3⋊D4
kernelC626D4D6⋊Dic3C6.D12Dic3⋊Dic3C2×C3⋊D12Dic3×C2×C6C6×C3⋊D4C2×C327D4C22×Dic3C2×C3⋊D4C3×Dic3C62C2×Dic3C22×S3C22×C6C3×C6Dic3C2×C6C2×C6C6C23C6C6C22C22C2C2
# reps111111111122312244441112122

Matrix representation of C626D4 in GL6(𝔽13)

0120000
110000
001000
000100
000073
0000106
,
100000
010000
0012100
0012000
0000120
0000012
,
100000
12120000
003700
0061000
000092
0000114
,
100000
12120000
0001200
0012000
0000120
000091

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,0,0,0,3,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,9,0,0,0,0,0,1] >;

C626D4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_6D_4
% in TeX

G:=Group("C6^2:6D4");
// GroupNames label

G:=SmallGroup(288,626);
// by ID

G=gap.SmallGroup(288,626);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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